Tropical geometry is a relatively recent field in mathematics created as a simplified model for certain problems in algebraic geometry. We introduce the definition of abstract and planar tropical curves as well as their properties, including combinatorial type and degree. We also talk about the moduli space, a geometric object that parameterizes all possible types of abstract or planar tropical curves subject to certain conditions. Our research focuses on the moduli spaces of planar tropical curves of genus one, arbitrary degree d and any number of marked, unbounded edges. We prove that these moduli spaces are connected.
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In this paper, we study the deformations of cyclic covers over a complete discrete valuation ring of equal characteristic p > 0. We first introduce the towers of Hurwitz trees notion, which arises from given towers of cyclic covers of a rigid disc. We then apply the Hurwitz tree technique to prove that one can always extend an equal characteristic deformation of a cyclic cover's subcover to a deformation of the whole cover. That shows the exact analog of the refined lifting conjecture for cyclic covers (conjectured by Saïdi) in equal characteristic is true.
Suppose φ is a Z/4-cover of a curve over an algebraically closed field k of characteristic 2, and φ 1 is its Z/2-sub-cover. Suppose, moreover, that Φ 1 is a lift of φ 1 to a complete discrete valuation ring R that is a finite extension of the ring of Witt vectors W (k) (hence in characteristic zero). We show that there exists a finite extension R ′ of R, and a lift Φ of φ to R ′ with a sub-cover isomorphic to Φ 1 ⊗ k R ′ . This gives the first non-trivial family of cyclic covers where Saïdi's refined lifting conjecture holds.
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